Liouville theorem for entire functions
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1: 1.9 Calculus of a Complex Variable
Liouville’s Theorem
…2: 15.16 Products
§15.16 Products
… ►where ${A}_{0}=1$ and ${A}_{s}$, $s=1,2,\mathrm{\dots}$, are defined by the generating function ►Generalized Legendre’s Relation
… ►For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).3: 18.39 Applications in the Physical Sciences
4: Errata
The following additions were made in Chapter 1:

§1.2
New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

§1.3
The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
 §1.4

§1.8
In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

§1.10
New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

§1.13
New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: SturmLiouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

§1.14(i)
Another form of Parseval’s formula, (1.14.7_5).

§1.16
We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

§1.17
Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

§1.18
An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).
The original constraint, $\mathrm{\Re}s>0$, was removed because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta \left(s\right){(s1)}^{1}$ is an entire function.
Suggested by John Harper.
The entire Section was replaced.